Elimination Method 2

Years

Year 10

Simultaneous Equations

Elimination Method

Elimination Method 2

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Question 1 of 5

1. Question
Solve the following simultaneous equations by elimination.

$3 x + y = 5$ $3x+y=5$

$x + y = 3$ $x+y=3$
- $x =$ $x=$ (1)
  
  $y =$ $y=$ (2)
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Elimination Method

$1)$ $1)$ make sure a variable has same coefficients on the 2 equations

$2)$ $2)$ add or subtract the equations so that one variable is cancelled

$3)$ $3)$ solve for the variable that remains

$4)$ $4)$ substitute known value to one of the equations to solve for the other variable

First, label the two equations $1$ $1$ and $2$ $2$ respectively.

$3 x + y$ $3x+y$ $=$ $=$ $5$ $5$ Equation $1$ $1$

$x + y$ $x+y$ $=$ $=$ $3$ $3$ Equation $2$ $2$

Next, subtract equation $2$ $2$ from equation $1$ $1$ .

$3 x + y$ $3x+y$ $=$ $=$ $5$ $5$

$-$ $-$ $(x + y)$ $(x+y)$ $=$ $=$ $3$ $3$

$2 x$ $2x$ $=$ $=$ $2$ $2$ $y - y$ $y-y$ cancels out

Solve for $x$ $x$ from the difference.

$2 x$ $2x$ $=$ $=$ $2$ $2$

$2 x$ $2x$ $\div 2$ $div2$ $=$ $=$ $2$ $2$ $\div 2$ $div2$ Divide both sides by $2$ $2$

$x$ $x$ $=$ $=$ $1$ $1$

Now, substitute the value of $x$ $x$ into any of the two equations.

$x$ $x$ $+ y$ $+y$ $=$ $=$ $3$ $3$ Equation $2$ $2$

$1$ $1$ $+ y$ $+y$ $=$ $=$ $3$ $3$ $x = 1$ $x=1$

$1 + y$ $1+y$ $- 1$ $-1$ $=$ $=$ $3$ $3$ $- 1$ $-1$ Subtract $1$ $1$ from both sides

$y$ $y$ $=$ $=$ $2$ $2$

$x = 1, y = 2$ $x=1, y =2$

Question 2 of 5

Solve the following simultaneous equations by elimination.

5 x + y = - 6

$5x+y=-6$

x + 2 y = 24

$x+2y=24$

$x =$ $x=$ (-4)

$y =$ $y=$ (14)

Question 3 of 5

3. Question
Solve the following simultaneous equations by elimination.

$3 a - 4 b = 24$ $3a-4b=24$

$4 a - 2 b = 12$ $4a-2b=12$
- $a =$ $a=$ (0)
  
  $b =$ $b=$ (-6)
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In the elimination method you either add or subtract the equations to get the value of $a$ $a$ and $b$ $b$

First, label the two equations $1$ $1$ and $2$ $2$ respectively.

$3 a - 4 b$ $3a-4b$ $=$ $=$ $24$ $24$ Equation $1$ $1$

$4 a - 2 b$ $4a-2b$ $=$ $=$ $12$ $12$ Equation $2$ $2$

Multiply Equation $2$ $2$ by $2$ $2$ .

$4 a - 2 b$ $4a-2b$ $=$ $=$ $12$ $12$

$(4 a - 2 b)$ $(4a-2b)$ $\times 2$ $xx 2$ $=$ $=$ $12$ $12$ $\times 2$ $xx 2$

$8 a - 4 b$ $8a-4b$ $=$ $=$ $24$ $24$ Simplify

Subtract equation $1$ $1$ from the transformed equation.

$8 a - 4 b$ $8a-4b$ $=$ $=$ $24$ $24$

$3 a - 4 b$ $3a-4b$ $=$ $=$ $24$ $24$

$5 a$ $5a$ $=$ $=$ $0$ $0$ $- 4 b - (- 4 b)$ $-4b-(-4b)$ cancels out

Solve for $a$ $a$ .

$5 a$ $5a$ $=$ $=$ $0$ $0$

$a$ $a$ $=$ $=$ $0$ $0$ Divide both sides by $5$ $5$

Now, substitute the value of $a$ $a$ into any of the two equations.

$3$ $3$ $a$ $a$ $- 4 b$ $-4b$ $=$ $=$ $24$ $24$ Equation $1$ $1$

$3$ $3$ $(0)$ $(0)$ $- 4 b$ $-4b$ $=$ $=$ $24$ $24$ $a = 0$ $a=0$

$- 4 b$ $-4b$ $=$ $=$ $24$ $24$

$b$ $b$ $=$ $=$ $- 6$ $-6$ Divide both sides by $- 4$ $-4$

$a = 0, b = - 6$ $a=0, b=-6$
Question 4 of 5

4. Question
Solve the following simultaneous equations by elimination.

$a - 5 b = 8$ $a-5b=8$

$2 a - 3 b = 9$ $2a-3b=9$
- $x =$ $x=$ (3)
  
  $y =$ $y=$ (-1)
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Elimination Method

$1)$ $1)$ make sure a variable has same coefficients on the 2 equations

$2)$ $2)$ add or subtract the equations so that one variable is cancelled

$3)$ $3)$ solve for the variable that remains

$4)$ $4)$ substitute known value to one of the equations to solve for the other variable

First, label the two equations $1$ $1$ and $2$ $2$ respectively.

$a - 5 b$ $a-5b$ $=$ $=$ $8$ $8$ Equation $1$ $1$

$2 a - 3 b$ $2a-3b$ $=$ $=$ $9$ $9$ Equation $2$ $2$

Next, multiply the values of equation $1$ $1$ by $2$ $2$ and label the product as equation $3$ $3$ .

$a - 5 b$ $a-5b$ $=$ $=$ $8$ $8$ Equation $1$ $1$

$(a - 5 b)$ $(a-5b)$ $\times 2$ $times2$ $=$ $=$ $8$ $8$ $\times 2$ $times2$ Multiply the values of both sides by $2$ $2$

$2 a - 10 b$ $2a-10b$ $=$ $=$ $16$ $16$ Equation $3$ $3$

Then, subtract equation $2$ $2$ from equation $3$ $3$ .

$2 a - 10 b$ $2a-10b$ $=$ $=$ $16$ $16$

$-$ $-$ $(2 a - 3 b)$ $(2a-3b)$ $=$ $=$ $9$ $9$

$- 7 b$ $-7b$ $=$ $=$ $7$ $7$ $2 a - 2 a$ $2a-2a$ cancels out

Solve for $b$ $b$ from the difference.

$- 7 b$ $-7b$ $=$ $=$ $7$ $7$

$- 7 b$ $-7b$ $\div (- 7)$ $div(-7)$ $=$ $=$ $7$ $7$ $\div (- 7)$ $div(-7)$ Divide both sides by $- 7$ $-7$

$b$ $b$ $=$ $=$ $- 1$ $-1$

Now, substitute the value of $b$ $b$ into any of the two equations.

$a - 5$ $a-5$ $b$ $b$ $=$ $=$ $8$ $8$ Equation $1$ $1$

$a - 5$ $a-5$ $(- 1)$ $(-1)$ $=$ $=$ $8$ $8$ $b = - 1$ $b=-1$

$a + 5$ $a+5$ $- 5$ $-5$ $=$ $=$ $8$ $8$ $- 5$ $-5$ Subtract $5$ $5$ from both sides

$a$ $a$ $=$ $=$ $3$ $3$

$a = 3, y = - 1$ $a=3, y =-1$

Question 5 of 5

Solve the following simultaneous equations by elimination.

3 x - 5 y = 11

$3x-5y=11$

2 x - y = 5

$2x-y=5$

$x =$ $x=$ (2)

$y =$ $y=$ (-1)

$10 x + 2 y$ $10x+2y$	$=$ $=$	$- 12$ $-12$
$x + 2 y$ $x+2y$	$=$ $=$	$24$ $24$

$9 x$ $9x$	$=$ $=$	$- 36$ $-36$	$2 y - 2 y$ $2y-2y$ cancels out

$10 x - 5 y$ $10x-5y$	$=$ $=$	$25$ $25$
$-$ $-$ $(3 x - 5 y)$ $(3x-5y)$	$=$ $=$	$11$ $11$

$7 x$ $7x$	$=$ $=$	$14$ $14$	$- 5 y - (- 5 y)$ $-5y-(-5y)$ cancels out

$3$ $3$ $x$ $x$ $- 5 y$ $-5y$	$=$ $=$	$11$ $11$	Equation $1$ $1$
$3$ $3$ $(2)$ $(2)$ $- 5 y$ $-5y$	$=$ $=$	$11$ $11$	$x = 2$ $x=2$
$6 - 5 y$ $6-5y$ $- 6$ $-6$	$=$ $=$	$11$ $11$ $- 6$ $-6$	Subtract $6$ $6$ from both sides
$- 5 y$ $-5y$ $\div (- 5)$ $div(-5)$	$=$ $=$	$5$ $5$ $\div (- 5)$ $div(-5)$	Divide both sides by $- 5$ $-5$
$y$ $y$	$=$ $=$	$- 1$ $-1$

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Quiz summary

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1. Question

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Elimination Method

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3. Question

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4. Question

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Elimination Method

5. Question

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Elimination Method

Quizzes

$3 x + y$ $3x+y$	$=$ $=$	$5$ $5$	Equation $1$ $1$
$x + y$ $x+y$	$=$ $=$	$3$ $3$	Equation $2$ $2$

$2 x$ $2x$	$=$ $=$	$2$ $2$
$2 x$ $2x$ $\div 2$ $div2$	$=$ $=$	$2$ $2$ $\div 2$ $div2$	Divide both sides by $2$ $2$
$x$ $x$	$=$ $=$	$1$ $1$

$x$ $x$ $+ y$ $+y$	$=$ $=$	$3$ $3$	Equation $2$ $2$
$1$ $1$ $+ y$ $+y$	$=$ $=$	$3$ $3$	$x = 1$ $x=1$
$1 + y$ $1+y$ $- 1$ $-1$	$=$ $=$	$3$ $3$ $- 1$ $-1$	Subtract $1$ $1$ from both sides
$y$ $y$	$=$ $=$	$2$ $2$

$5 x + y$ $5x+y$	$=$ $=$	$- 6$ $-6$	Equation $1$ $1$
$x + 2 y$ $x+2y$	$=$ $=$	$24$ $24$	Equation $2$ $2$

$x$ $x$ $+ 2 y$ $+2y$	$=$ $=$	$24$ $24$	Equation $2$ $2$
$- 4$ $-4$ $+ 2 y$ $+2y$	$=$ $=$	$24$ $24$	$x = - 4$ $x=-4$
$- 4 + 2 y$ $-4+2y$ $+ 4$ $+4$	$=$ $=$	$24$ $24$ $+ 4$ $+4$	Add $4$ $4$ to both sides
$2 y$ $2y$	$=$ $=$	$28$ $28$
$y$ $y$	$=$ $=$	$14$ $14$	Divide both sides by $2$ $2$

$3 a - 4 b$ $3a-4b$	$=$ $=$	$24$ $24$	Equation $1$ $1$
$4 a - 2 b$ $4a-2b$	$=$ $=$	$12$ $12$	Equation $2$ $2$

$8 a - 4 b$ $8a-4b$	$=$ $=$	$24$ $24$
$3 a - 4 b$ $3a-4b$	$=$ $=$	$24$ $24$

$5 a$ $5a$	$=$ $=$	$0$ $0$	$- 4 b - (- 4 b)$ $-4b-(-4b)$ cancels out

$3$ $3$ $a$ $a$ $- 4 b$ $-4b$	$=$ $=$	$24$ $24$	Equation $1$ $1$
$3$ $3$ $(0)$ $(0)$ $- 4 b$ $-4b$	$=$ $=$	$24$ $24$	$a = 0$ $a=0$
$- 4 b$ $-4b$	$=$ $=$	$24$ $24$
$b$ $b$	$=$ $=$	$- 6$ $-6$	Divide both sides by $- 4$ $-4$

$a - 5 b$ $a-5b$	$=$ $=$	$8$ $8$	Equation $1$ $1$
$2 a - 3 b$ $2a-3b$	$=$ $=$	$9$ $9$	Equation $2$ $2$

$2 a - 10 b$ $2a-10b$	$=$ $=$	$16$ $16$
$-$ $-$ $(2 a - 3 b)$ $(2a-3b)$	$=$ $=$	$9$ $9$

$- 7 b$ $-7b$	$=$ $=$	$7$ $7$	$2 a - 2 a$ $2a-2a$ cancels out

$a - 5$ $a-5$ $b$ $b$	$=$ $=$	$8$ $8$	Equation $1$ $1$
$a - 5$ $a-5$ $(- 1)$ $(-1)$	$=$ $=$	$8$ $8$	$b = - 1$ $b=-1$
$a + 5$ $a+5$ $- 5$ $-5$	$=$ $=$	$8$ $8$ $- 5$ $-5$	Subtract $5$ $5$ from both sides
$a$ $a$	$=$ $=$	$3$ $3$

$3 x - 5 y$ $3x-5y$	$=$ $=$	$11$ $11$	Equation $1$ $1$
$2 x - y$ $2x-y$	$=$ $=$	$5$ $5$	Equation $2$ $2$